Why does my car sometimes prevent me from shifting gears, despite the clutch pedal being pressed down fully?

In the beginning, this only happened at stoplights. While waiting for the light to turn, I would try to shift back into first. Yes, the clutch is always pressed down all the way. But the shifter wouldn't go into first all the way. I would have to bring it back to neutral again and try again, until it went all the way into gear. This usually took one or two tries, but thankfully I was stopped at the light.

But today this happened as I was trying to shift into second. I thought I had damaged second gear since I had made a rough downshift a few minutes earlier. I pulled over and went back into the road, but once again the car failed to shift into second. I tried a third time and it engaged.

What causes a shifter to refuse or fail to fully engage a gear? Do you think this is a problem with my car, or how I am driving?

Thank you.